Related papers: On exterior differential systems involving differe…

The thesis studies Frobenius-type theorems in non-smooth settings. We extend the definition of involutivity to non-Lipschitz subbundles using generalized functions. We prove the real Frobenius Theorem with sharp regularity on log-Lipschitz…

We consider nonlinear parabolic evolution equations of the form $\partial_{t}u=F(t,x,Du,D^{2}u) $, subject to noise of the form $H(x,Du) \circ dB$ where $H$ is linear in $Du$ and $\circ dB$ denotes the Stratonovich differential of a…

Within the context of rough path analysis via fractional calculus, we show how variability can be used to prove the existence of integrals with respect to H\"older continuous multiplicative functionals in the case of Lipschitz coefficients…

Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued H\"{o}lder continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and $f$ is a continuously…

We extend the theory of exterior differential systems from manifolds and their tangent bundles to Lie algebroids. In particular, we define the concept of an integral manifold of such an exterior differential system. We support our…

In this article we extend the framework of rough paths to processes of variable H\"older exponent or variable order paths. We show how a class of multiple discrete delay differential equations driven by signals of variable order are…

We prove a general existence theorem for nonlinear partial differential systems of any order in one complex variable. A special case of first order contains a well-known theorem of Nijenhuis and Woolf concerning local existence of…

We extend the Lyapunov function technique, a fundamental tool for investigating asymptotic stability and existence of attractors for ordinary differential equations, by introducing the notion of a {\it strong Lyapunov function} for an…

We apply Frobenius integrability theorem in the search of invariants for one-dimensional Hamiltonian systems with a time-dependent potential. We obtain several classes of potential functions for which Frobenius theorem assures the existence…

The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well…

Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian…

We study weak solutions ${\bf v}:U\times (0,T)\rightarrow \mathbb{R}^m$ of the nonlinear parabolic system $$ D\psi({\bf v}_t)=\text{div}DF(D{\bf v}), $$ where $\psi$ and $F$ are convex functions. This is a prototype for more general doubly…

We briefly recall a fundamental exterior differential system introduced by the author and then apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemaniann 3-manifolds. The system…

Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…

We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat…

In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical…

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…

Rough paths theory allows for a pathwise theory of solutions to differential equations driven by highly irregular signals. The fundamental observation of rough paths theory is that if one can define "iterated integrals" above a signal, then…

This article introduces the splitting method to systems responding to rough paths as external stimuli. The focus is on nonlinear partial differential equations with rough noise but we also cover rough differential equations. Applications to…

The bases of the theory of integrals for multidimensional differential systems are stated. The integral equivalence of total differential systems, linear homogeneous systems of partial differential equations, and Pfaff systems of equations…